Rare weapons are high rarity weapons that often have pretty strong effects. Higher rarity weapons will naturally have higher base stats, even if they are on the same level, but the main selling point of rare weapons are their secondary stat and main effect. At rarity 5, weapons will have unconditional stat boost in addition to their main effect.
How to get rare weapons
If you want to bump up your attack, you're going to need strong weapons. This is important because artifact sub stats that increases your atk by a percentage is based on your character's base stats + weapon attack. Higher rarity weapons will naturally have higher attack, so it's what you're going to want to aim for.
The first option is to craft a weapon. You can do this by going to the blacksmiths in the city. There are no differences between the blacksmith in Mondstadt and Liyue.
The blacksmith can craft 2 different weapons of each type, a prototype version, or a non-prototype version. They have different effects, and also a different secondary effect. Craft the one that you think will be useful for you.
In order to craft the weapons, you need to get the prototype material, and also some ores. You can get the prototype material from 2 places.
The first one is by buying them from the shop with the diamond icon in Mondstadt/Liyue, and it costs Anemo/Geo sigils, depending on which city you talk to.
The second way is to fight the 2 available bosses that costs 60 resin, and they are Dvalin, and Andrius. The drops are random though, so getting them is not guaranteed. You need to be at least Adventure Rank 21 to fight Andrius, and 25 for Dvalin. Check the Adventure Handbook to view their location and drops.
The second way of getting good weapons is through the wish/gacha system. The limited rate up banners usually use Intertwined Fates, but you can also get some rarity 4/5 weapons from the standard banners that uses Acquainted Fates.
The other option is through the Battle Pass. You unlock the Battle Pass when you reach Adventure Rank 20, and you get to pick a Battle Pass weapon of your choice when you reach BP level 30. You must upgrade your Battle Pass though, and this costs $10.
The prototype version, or the non prototype version? Wolf's Gravestone, or Skyward Pride? Since the weapons themselves have varying effects, it's important to know if the weapon will fit your playstyle. In general, it's better to go for weapons that increases attack, but if you already have a main damage dealer, then you can equip supporting type weapons that have secondary stats that increase energy recharge, or elemental mastery instead. "Prototype Aminus or Whiteblind?" is a common question people like to ask to compare. Both are not bad options but if you are crafting the weapon for your main damage dealer, like Diluc, then go for Aminus since it increases ATK as the secondary stat, Whiteblind can be a good option for characters like Noelle since it increases DEF as the secondary. Secondary stats that increase CRIT Rate or Crit damage can be good too if your artifact stats supports it.